Fig. 1. CROW configurations. N identical Kerr ring resonators are linked in sequence. Input field directions ensure that each resonator’s circulating field travels only in a singular direction. Note that systems with odd and even numbers of resonators require differing input directions for the end resonators; see faded resonators of the figure. (a) “Input to end” CROW: inputs are provided only to the end resonators. (b) “Input to all” CROW: inputs are connected to all resonators. Input (output) directions are shown by red (purple) arrows.

Fig. 2. Evolutions of optical intensities in N=3 “input to end” CROW system. Panels (a)–(c) show the evolutions of the field intensities in different resonators as a function of input power. The field intensities in the end resonators are depicted in blue and black, whereas the field intensity in the middle resonator is depicted in red. For (a), ζ=0.5, j=1. A field intensity crossing point appears in (a). Before the crossing point, the field intensity in the middle resonator is higher than the fields in the end resonators and beyond this point, the end resonator fields become more intense than the middle resonator field. Panel (b) depicts spontaneous symmetry breaking of the end resonator light field intensities for ζ=5, j=2. Symmetry unbroken and symmetry broken oscillations are depicted in the upper (ζ=3, j=2) and lower (ζ=5, j=2.5) panels of (c), respectively. In this and all successive figures, the dotted lines stand for the positions of the maxima and the minima of the oscillations, and the shaded regions in between highlight the span of the oscillations. Upper and lower panels of (d) are examples of symmetry unbroken and complete symmetry broken oscillations, respectively. Used parameters: |f|2=22, ζ=3, j=2 [(d) upper panel], |f|2=37.79, ζ=5, j=2.5 [(d) lower panel]. Time step for integration is 0.005.

Fig. 3. Circulating field intensities, Ψn(=|ψn|2), against input power, |f|2, for an N=3 “input to end” CROW system. For detuning ζ=5 and inter-resonator coupling j=2, we present in (a), (b) the analytical solutions to Eq. (1) for the fields circulating the middle Ψ2 and end resonators Ψ1,3, respectively. In panels (c), (d) and panels (e), (f), respectively, we display the results of numerical integrations of Eq. (1) for stepwise increasing and stepwise decreasing values of the input field intensity. In all panels, different “relationships” of solutions are colored accordingly for visual benefit; by this we mean that when fields Ψ1,3 are on the green solution line, this means that Ψ2 is also on its own respective green solution line. These results are discussed thoroughly in the main text, but we highlight the possibility of end-resonator-symmetric solutions (red) and two distinct end-resonator-symmetry-broken solutions (green and blue).

Fig. 4. Occurrence of SSB in N=3 CROW system. The left side of the dashed line corresponds to a scan with an input power of 22.20, whereas the right side of the scan corresponds to a scan with an input power of 22.42. It can be seen that starting from a stable “black-blue symmetric” state, the system after the change in input power, goes to a “black-blue symmetry broken” state after a small oscillatory evolution. The origin of the oscillations can be described from the linear stability analysis presented in Appendices C and D. The blue and black lines represent the circulating field intensities in the end resonators, whereas the red line depicts the field in the middle resonator. The vertical dashed black line marks the point where the input power is increased. In each case, the detuning is five. Time step for integration is 0.005.

Fig. 5. Evolutions of optical intensities in (a)–(c) N=5 and (d)–(f) N=10 “input to end” CROW systems. Panel (a) and (b)-upper panel show the evolutions of the field intensities in different resonators as a function of input power. The end resonator field intensities are depicted in cyan and blue. In (a), the end resonator field intensities and the neighboring resonator’s field intensities (depicted in red and green) display spontaneous symmetry breakings. In (b), the field intensities within coupling-wise symmetric resonators always remain symmetric and they display oscillations. Overlapping (lower-left panel) and non-overlapping (lower-right panel) oscillations are observed. In (c), we demonstrate three of the possible light intensity distribution conditions in the resonators with bright-red (dark-red) referring to bright (dark) resonators (left configuration: ζ=3.62, J=1, |f|2=59.18; middle configuration: ζ=3.62, J=1, |f|2=37.88; and right configuration: ζ=3, J=2, |f|2=112.99). Input power scans for N=10 CROW systems are depicted in (d) and (e)-upper panel. All the symmetric field pairs undergo SSBs (in intensity) in (d) and upper panel of (e), followed by bistability jumps and symmetry broken oscillations. Moreover, symmetry unbroken oscillations are observed in the upper panel of (e). In the lower panel of (e), three examples of oscillations of field intensities in time are depicted from the different oscillatory regions as shown in the corresponding upper panel. Panel (f) presents three possible light distribution conditions (left configuration: ζ=1.5, J=4, |f|2=37.88; middle configuration: ζ=1, J=6, |f|2=270.67; and right configuration: ζ=1, J=6, |f|2=88.38). Used parameters: (a) ζ=3.62 and j=1, (b) ζ=3 and j=2, (d) ζ=1.5 and j=4, and (e) ζ=1 and j=6. Time step for integration is 0.005.

Fig. 6. Colormap of different symmetry breaking conditions as a function of input power and detuning for (a) N=3, (b) N=5, (c) N=10, and (d) N=20 “input to end” CROW systems. Purple regions stand for symmetry unbroken case and the yellow stands for completely asymmetric case. The colors in between (different shades of green) stand for different levels of symmetry breakings in the systems. The watershed portions with stripes in each panel display the regions of oscillations. In all cases, the scans are done from lower to higher input powers for certain detuning values. In other words, Eq. (1) is scanned for all detuning values starting from zero input power. For each input power, the initial values of the field amplitudes are selected to be the steady state value of the last step (smaller input power). The scans for all detunings are done in parallel. For the scans j=2  (a), 1 (b), 3.5 (c), and 3 (d).

Fig. 7. Evolutions of optical intensities in “input to all” CROW systems for (b) N=4, (c) N=5, (e) N=10, and (f) N=50. Panel (a) depicts the schematic for a system with N=4. Panel (b) presents the evolutions of the field intensities in an N=4 CROW system with increasing input power for ζ=1.5 and j=1. Oscillations, bistability jumps, and SSBs are observed sequentially in the system with increasing input power. Panel (c) shows the evolutions of the field intensities in an N=5 CROW system with increasing input power for ζ=2.5 and j=2. Panel (d) displays the Poincaré section plot of (c). In the region where dots of all colors look randomly scattered, chaos is observed. Panel (e) and the upper panel of (f) present evolutions of the field intensities for N=10 (ζ=1.5, j=1) and N=50 (ζ=1.5, j=1) CROW systems, respectively. In both cases, the fields within coupling-wise symmetric resonators break their symmetry with increasing input power; afterwards, they form two bunches, one at high power and the other one at low power. Thereafter in panel (e), by a bistability jump, the field intensities form two inverse bifurcation structures. In the upper panel of (f), several bistability jumps can be observed, where different field intensities group and regroup at different jumps. Finally, two inverse bifurcation structures form. The lower panel of (f) demonstrates the bright (yellow)-dark (purple) conditions of different resonators for different input powers.

Fig. 8. Field intensity crossings for N=3 “input to end” CROW systems. Panels (a)–(c) show input power scan of resonator light intensities for CROW systems with N=3. In panel (a) the end resonator field intensities (shown in black) do not cross the field intensity in the middle resonator (shown in red). The middle resonator field intensity crosses end resonator field intensities once in panel (b) and twice in panel (c). Panel (d) shows the ratio of field intensities circulating in end resonators and middle resonator. Brown line shows the ratio for (b), and green line shows the ratio for (c). Panel (d) demonstrates the potential for relative power distributions among the resonators in N=3 CROW systems. Used parameters: (a) ζ=0.5, j=0.1, (b) ζ=0.5, j=1, and (c) ζ=1.66, j=0.6.

Fig. 9. SSB mechanism in N=3 CROW system. Panels (a) and (b) show real and imaginary parts of eigenvalues of the Jacobian matrix of the system as given by Eq. (C2). The shaded region in panel (a) depicts the region where y-axis is above zero. When the real part of any of the eigenvalues enters the shaded region, the system becomes unstable. The dashed line (which is at the same place as the dashed line in Fig. 4 of the main manuscript) depicts the input power value for which the real part of one eigenvalue of the system goes above zero. The SSB between the circulating intensities in the end resonators occurs at this point.

Fig. 10. Switchings in “input to end” CROW systems. Periodic switchings of the fields within coupling-wise symmetric resonators for (a), (b) N=5, (c), (d) N=10, (e)–(h) N=20 are depicted. Perfect sinusoidal switchings (a), (c), (e), (g) are confirmed by the complete overlaps of the phase space plots (b), (d), (f), (h). Used parameters: (a) |f|2=108.34, ζ=3, j=1, (c) |f|2=39.18, ζ=4.2, j=3.5, (e) |f|2=67.53, ζ=4, j=3, (g) |f|2=120.05, ζ=4, j=3. Time step for integration is 0.005.

Fig. 11. Oscillations in “input to all” CROW systems. (a) Input power scan of resonator light intensities for CROW system with N=4, ζ=1.5, and j=1. (b) Evolutions of field intensities over time in the oscillatory region [shaded region in (a)] with |f|2=5.68. (c) Input power scan of resonator light intensities for CROW system with N=5, ζ=2,5, and j=2. (d) and (e) show oscillations with maintained symmetry between fields within resonators with symmetric coupling conditions, whereas (f) shows switching oscillations between them. (g) depicts five-field chaos. (h) shows near-switching oscillations. |f|2=12.93  (d),20.20  (e),24.44  (f),29.09  (g),34.14  (h). Time step for integration is 0.005.

Fig. 12. Effects of fabrication-induced asymmetries in “input to end” CROW systems. Panels (a)–(c) show evolutions of circulating powers in different resonators with increasing input power in N=10 “input to end” CROW systems. The detunings for the different resonators are assigned from a Gaussian distribution of mean 1 and standard deviation 0% (a), 5% (b), and 10% (c). Time step for integration is 0.005.

Fig. 13. Nonlinear enhancement of asymmetries in “input to end” CROW systems with nonidentical resonators. Evolutions of circulating powers in resonators (a) 1 and 10 and (b) 3 and 8 with increasing input power in N=10 “input to end” CROW system are depicted. With increasing variations of inherent inter-resonator detuning, SSB is substituted by nonlinear enhancement of inter-resonator optical power difference. The detunings for the different resonators are assigned from a Gaussian distribution of mean 1 and standard deviation 0% (dotted lines), 1% (dashed lines), 5% (dashed-dotted lines), and 10% (solid lines) of its mean. Inter-resonator coupling is set to j=6. Field intensities display oscillatory responses in the shaded regions. Time step for integration is 0.005.